| L(s) = 1 | + 0.618·2-s + 0.618·3-s − 0.618·4-s + 0.381·6-s − 1.61·7-s − 8-s − 0.618·9-s + 11-s − 0.381·12-s + 13-s − 1.00·14-s − 0.381·18-s + 0.618·19-s − 1.00·21-s + 0.618·22-s − 1.61·23-s − 0.618·24-s + 25-s + 0.618·26-s − 27-s + 0.999·28-s + 0.999·32-s + 0.618·33-s + 0.381·36-s + 0.381·38-s + 0.618·39-s − 1.61·41-s + ⋯ |
| L(s) = 1 | + 0.618·2-s + 0.618·3-s − 0.618·4-s + 0.381·6-s − 1.61·7-s − 8-s − 0.618·9-s + 11-s − 0.381·12-s + 13-s − 1.00·14-s − 0.381·18-s + 0.618·19-s − 1.00·21-s + 0.618·22-s − 1.61·23-s − 0.618·24-s + 25-s + 0.618·26-s − 27-s + 0.999·28-s + 0.999·32-s + 0.618·33-s + 0.381·36-s + 0.381·38-s + 0.618·39-s − 1.61·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7608334685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7608334685\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 0.618T + T^{2} \) |
| 3 | \( 1 - 0.618T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.61T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 0.618T + T^{2} \) |
| 23 | \( 1 + 1.61T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.61T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.618T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.618T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.61T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.58960547893136761773464901160, −12.59167174571363529386519262535, −11.68668611219368486407903566608, −10.01498252321731105275878949683, −9.173437165438516524913779804257, −8.425902717004282755503329240086, −6.60936187943298278927509634323, −5.73504655678137618143092519957, −3.89660235366871617514810629621, −3.16035529886055676861741520657,
3.16035529886055676861741520657, 3.89660235366871617514810629621, 5.73504655678137618143092519957, 6.60936187943298278927509634323, 8.425902717004282755503329240086, 9.173437165438516524913779804257, 10.01498252321731105275878949683, 11.68668611219368486407903566608, 12.59167174571363529386519262535, 13.58960547893136761773464901160
